Worksheets · Calculus

Graphing the derivative worksheet

Where f turns, its derivative crosses zero. Where f rises, the derivative sits above the axis, and where it falls, below. Reading those three things off the graph is enough to draw f' without a formula.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing the Derivative

Date Period

Use the graph of f to sketch the graph asked for.

  1. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  2. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  3. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  4. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  5. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  6. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  7. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  8. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  9. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  10. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  11. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  12. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  13. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  14. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  15. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  16. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  17. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  18. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010
  19. The graph of f is shown. Sketch the graph of f′.y = f(x)y = f′(x)−10−10−5−500551010−10−10−5−500551010
  20. The graph of f is shown. Sketch the graph of f′′.y = f(x)y = f″(x)−10−10−5−500551010−10−10−5−500551010

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Graphing the DerivativeAnswer keyVersion 1

Date Period

Use the graph of f to sketch the graph asked for.

  1. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=32Positive on (−∞,32)Negative on (32,∞)
  2. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=−5 and x=−1Positive on (−5,−1)Negative on (−∞,−5)∪(−1,∞)
  3. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=−1Positive on (−∞,−1)Negative on (−1,∞)
  4. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=−4 and x=0Positive on (−4,0)Negative on (−∞,−4)∪(0,∞)
  5. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=−2 and x=2Positive on (−2,2)Negative on (−∞,−2)∪(2,∞)
  6. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=2Positive on (2,∞)Negative on (−∞,2)
  7. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=−52Positive on (−∞,−52)Negative on (−52,∞)
  8. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=0 and x=4Positive on (0,4)Negative on (−∞,0)∪(4,∞)
  9. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=−32Positive on (−32,∞)Negative on (−∞,−32)
  10. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=−2 and x=3Positive on (−2,3)Negative on (−∞,−2)∪(3,∞)
  11. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=−3 and x=1Positive on (−∞,−3)∪(1,∞)Negative on (−3,1)
  12. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=52Positive on (52,∞)Negative on (−∞,52)
  13. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=−12Positive on (−12,∞)Negative on (−∞,−12)
  14. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=−1 and x=2Positive on (−∞,−1)∪(2,∞)Negative on (−1,2)
  15. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=−5 and x=0Positive on (−∞,−5)∪(0,∞)Negative on (−5,0)
  16. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=12Positive on (−∞,12)Negative on (12,∞)
  17. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=2 and x=5Positive on (−∞,2)∪(5,∞)Negative on (2,5)
  18. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=2Positive on (−∞,2)Negative on (2,∞)
  19. The graph of f is shown. Sketch the graph of f′.−10−10−5−500551010f′=0 at x=−2 and x=2Positive on (−∞,−2)∪(2,∞)Negative on (−2,2)
  20. The graph of f is shown. Sketch the graph of f′′.−10−10−5−500551010f′′=0 at x=−72Positive on (−∞,−72)Negative on (−72,∞)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

f rises to a high point at x=−1, falls to a low point at x=2, then rises. Sketch f′.

  1. f' is zero at the two turning points, x = -1 and x = 2.
  2. f rises before -1 and after 2, so f' is above the axis there.
  3. f falls between -1 and 2, so f' is below the axis there: an upward parabola through -1 and 2.

The answer is f′=0 at x=−1 and x=2.

Where students go wrong

Copying the shape of f. The derivative is the slope, not the height: a high point of f is a zero of f', not a high point of it.

Also called sketching the derivative, graph of f prime, graphing f prime from f, derivative graphs, comparing a function and its derivatives or f, f' and f''.

Questions about these worksheets

Yes. Every graphing the derivative sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the graphing the derivative questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Copying the shape of f. The derivative is the slope, not the height: a high point of f is a zero of f', not a high point of it.